This is a conditional clause. The usual construction of the sentence would be :
If she had done so she would have concluded that my compliment was nonsense.
Main clause: She would have concluded that my compliment was nonsense.
Subordinate clause: If she had done so.
In literary English when the auxiliary verb of the subordinate "if clause" is "had", "were", or "should" it is possible to have this auxiliary verb at the beginning of the subordinate clause and omit "if".
Note: you could not have "have" (as stated in your question) because conditional can only be expressed with "had", and not "have")
If it were true, I would know.
→ Were it true, I would know.
If you had told me you were coming, I would have baked a cake.
→ Had you told me you were coming, I would have baked a cake.
When the subordinate clause does not have "were" or "had" it is still possible to introduce "were" or "should" and have auxiliary verb/subject inversion by introducing an auxiliary verb:
- If you arrive early, I'll come and pick you up.
→ Should you arrive early, I would pick you up.
→ Were you to arrive early, I could pick you up.